SECT. 1.] TO POSITION IN THE ORBIT. 33 



32. 



Let us proceed now to the application of these principles to the most useful 

 of the operations above explained. 



I. If (f and E are supposed to be exactly given in using the formula VII., 

 article 8, for computing the true anomaly from the eccentric anomaly in the 

 elliptic motion, then in log tan (45 (f) and log tan i E, the error w may be 

 committed, and thus in the difference = log tan i v, the error 2w; therefore the 

 greatest error in the determination of the angle v will be 



3 at di v 3 w sin v 



d log tan I v 2 1 



I. denoting the modulus of the logarithms used in this calculation. The error, 

 therefore, to which the true anomaly v is liable, expressed in seconds, becomes 



^Ap 206265 = 0&quot;.0712 sin v, 



if Brigg s logarithms to seven places of decimals are employed, so that we may 

 be assured of the value of v within 0&quot;.07 ; if smaller tables to five places only, are 

 used, the error may amount to 7&quot;. 12. 



II. If e cos E is computed by means of logarithms, an error may be committed 

 to the extent of 



3 ta e cos E 



~T 



therefore the quantity 



1 e cos E. or - , 



a * 



will be liable to the same error. In computing, accordingly, the logarithm of this 

 quantity, the error may amount to (1 -)- &amp;lt;?) w &amp;gt; denoting by d the quantity 



3 e cos E 

 1 ecosJS 



taken positively : the possible error in log r goes up to the same limit, log a being 

 assumed to be correctly given. If the eccentricity is small, the quantity d is 

 always confined within narrow limits; but when e differs but little from 1, 

 1 e cos E remains very small as long as E is small ; consequently, 8 may 



5 



